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18
Foundations for decision problems in separation logic with general inductive predicates
 In Proc. FoSSaCS 2014
, 2014
"... Abstract. We establish foundational results on the computational complexity of deciding entailment in Separation Logic with general inductive predicates whose underlying base language allows for pure formulas, pointers and existentially quantified variables. We show that entailment is in general u ..."
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Abstract. We establish foundational results on the computational complexity of deciding entailment in Separation Logic with general inductive predicates whose underlying base language allows for pure formulas, pointers and existentially quantified variables. We show that entailment is in general undecidable, and ExpTimehard in a fragment recently shown to be decidable by Iosif et al. Moreover, entailment in the base language is ΠP2complete, the upper bound even holds in the presence of list predicates. We additionally show that entailment in essentially any fragment of Separation Logic allowing for general inductive predicates is intractable even when strong syntactic restrictions are imposed. 1
Parametric completeness for separation theories
, 2013
"... In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation m ..."
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Cited by 9 (2 self)
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In this paper, we close the logical gap between provability in the logic BBI, which is the propositional basis for separation logic, and validity in an intended class of separation models, as employed in applications of separation logic such as program verification. An intended class of separation models is usually specified by a collection of axioms describing the specific model properties that are expected to hold, which we call a separation theory. Our main contributions are as follows. First, we show that several typical properties of separation theories are not definable in BBI. Second, we show that these properties become definable in a suitable hybrid extension of BBI, obtained by adding a theory of naming to BBI in the same way that hybrid logic extends normal modal logic. The binderfree extension HyBBI captures most of the properties we consider, and the full extension HyBBI(↓) with the usual ↓ binder of hybrid logic covers all these properties. Third, we present an axiomatic proof system for our hybrid logic whose extension with any set of “pure ” axioms is sound and complete with respect to the models satisfying those axioms. As a corollary of this general result, we obtain, in a parametric manner, a sound and complete axiomatic proof system for any separation theory from our considered class. To the best of our knowledge, this class includes all separation theories appearing in the published literature. Categories and Subject Descriptors F.3.1 [Logics and Mean
Bunched Logics Displayed
, 2010
"... We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show how to constrain app ..."
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Cited by 8 (4 self)
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We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show how to constrain applications of displayequivalence in our calculi in such a way that an exhaustive proof search needbe only finitely branching, and establish a full deduction theorem for the bunched logics with classical additives, BBI and CBI. We also show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic are very unlikely to exist.
Automated cyclic entailment proofs in separation logic
 In CADE’11
, 2011
"... Abstract. We present a general automated proof procedure, based upon cyclic proof, for inductive entailments in separation logic. Our procedure has been implemented via a deep embedding of cyclic proofs in the HOL Light theorem prover. Experiments show that our mechanism is able to prove a number of ..."
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Cited by 7 (1 self)
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Abstract. We present a general automated proof procedure, based upon cyclic proof, for inductive entailments in separation logic. Our procedure has been implemented via a deep embedding of cyclic proofs in the HOL Light theorem prover. Experiments show that our mechanism is able to prove a number of nontrivial entailments involving inductive predicates. 1
CLASSICAL BI: ITS SEMANTICS AND PROOF THEORY
"... Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including ..."
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Cited by 7 (5 self)
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Abstract. We present Classical BI (CBI), a new addition to the family of bunched logics which originates in O’Hearn and Pym’s logic of bunched implications BI. CBI differs from existing bunched logics in that its multiplicative connectives behave classically rather than intuitionistically (including in particular a multiplicative version of classical negation). At the semantic level, CBIformulas have the normal bunched logic reading as declarative statements about resources, but its resource models necessarily feature more structure than those for other bunched logics; principally, they satisfy the requirement that every resource has a unique dual. At the prooftheoretic level, a very natural formalism for CBI is provided by a display calculus à la Belnap, which can be seen as a generalisation of the bunched sequent calculus for BI. In this paper we formulate the aforementioned model theory and proof theory for CBI, and prove some fundamental results about the logic, most notably completeness of the proof theory with respect to the semantics. 1.
A unified display proof theory for bunched logic
 in Proceedings of MFPS26
"... We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show that the standard sequ ..."
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Cited by 7 (3 self)
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We formulate a unified display calculus proof theory for the four principal varieties of bunched logic by combining display calculi for their component logics. Our calculi satisfy cutelimination, and are sound and complete with respect to their standard presentations. We show that the standard sequent calculus for BI can be seen as a reformulation of its display calculus, and argue that analogous sequent calculi for the other varieties of bunched logic seem very unlikely to exist.
A Theorem Prover for Boolean BI
"... While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we d ..."
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Cited by 6 (0 self)
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While separation logic is acknowledged as an enabling technology for largescale program verification, most of the existing verification tools use only a fragment of separation logic that excludes separating implication. As the first step towards a verification tool using full separation logic, we develop a nested sequent calculus for Boolean BI (Bunched Implications), the underlying theory of separation logic, as well as a theorem prover based on it. A salient feature of our nested sequent calculus is that its sequent may have not only smaller child sequents but also multiple parent sequents, thus producing a graph structure of sequents instead of a tree structure. Our theorem prover is based on backward search in a refinement of the nested sequent calculus in which weakening and contraction are built into all the inference rules. We explain the details of designing our theorem prover and provide empirical evidence of its practicality.
Proof search for propositional abstract separation logics via labelled sequents
 In POPL’14. ACM
, 2014
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Expressive Completeness of Separation Logic With Two Variables and No Separating Conjunction ∗
"... We show that firstorder separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak secondorder logic, substantially sharpening a previous result. Capturing weak secondorder logic with such a restricted form ..."
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Cited by 4 (0 self)
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We show that firstorder separation logic with one record field restricted to two variables and the separating implication (no separating conjunction) is as expressive as weak secondorder logic, substantially sharpening a previous result. Capturing weak secondorder logic with such a restricted form of separation logic requires substantial updates to known proof techniques. We develop these, and as a byproduct identify the smallest fragment of separation logic known to be undecidable: firstorder separation logic with one record field, two variables, and no separating conjunction.
Separation logic with one quantified variable. arXiv
, 2014
"... Abstract. We investigate firstorder separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is PSPACEcomplete for the propositional fragment. We show that ..."
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Abstract. We investigate firstorder separation logic with one record field restricted to a unique quantified variable (1SL1). Undecidability is known when the number of quantified variables is unbounded and the satisfiability problem is PSPACEcomplete for the propositional fragment. We show that the satisfiability problem for 1SL1 is PSPACEcomplete and we characterize its expressive power by showing that every formula is equivalent to a Boolean combination of atomic properties. This contributes to our understanding of fragments of firstorder separation logic that can specify properties about the memory heap of programs with singlylinked lists. When the number of program variables is fixed, the complexity drops to polynomial time. All the fragments we consider contain the magic wand operator and firstorder quantification over a single variable. 1